Min-max minimal disks with free boundary in Riemannian manifolds

Abstract

We establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for the Plateau problem of minimal disks, which can be used to generalize the famous work by Morse–Tompkins and Shiffman on minimal surfaces in ${ℝ}^n$ to the Riemannian setting.

More precisely, we generalize, to the free boundary setting, the min-max construction of minimal surfaces using harmonic replacement introduced by Colding–Minicozzi. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit $2$–disk in ${ℝ}^2$ into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit $2$–disk proved by Colding and Minicozzi.